a new Electrical Theorem. . 449 



and, where a point of junction is encountered, each to become 

 a seat of the same electromotive force in each of the newly 

 encountered bars (of course leaving the resistance of the 

 source behind), then the disposition at any moment is equi- 

 valent to that at any other moment, and therefore to the 

 original disposition. [Of course the direction of the electro- 

 motive force must be carefully maintained the same : if it is 

 towards the knot before crossing it, it must be away from the 

 knot after passing it.] The proof need only refer to the 

 passing of a knot point, for no one will doubt that if the 

 sources only move in an unbranching portion of the conductor 

 the currents in different parts of the system will remain the 

 same. 



Let therefore the source e approach the point A at which its 

 path splits into n other ways. In each of the n bars suppose 

 a source e inserted as directed, then these n alone must be 

 equivalent to the single source before reaching A ; for if the 

 n sources are reversed, the current due to these sources in 

 every portion of the system is reduced to zero. The reversed 

 n sources would therefore alone produce currents in the system 

 equal numerically, but opposite in direction, to those produced 

 by the single source. Hence it follows that the n sources 

 (not reversed) will produce the same current as the single 

 source. 



The principle of the superposition of currents enables us to 

 apply this result to each source of the system, and therefore 

 to prove the truth of the theorem in its complete generality. 



In equivalent systems, since the current in every part 

 remains the same, the total power expended remaius the 

 same ; and equivalent systems might have been denned as 

 those which produce the same expenditure of power in each 

 part ; and therefore the total power of the sources remains 

 the same. 



From the point of view of KirchhofFs theorem %e = S.rc 

 for any closed path in a network, the above general theorem 

 may seem to some minds even plainer and more easily proved 

 than on the method of demonstration which I have employed. 



